Ingeniería Biomédica
2025-07-23
Importante
The digital filter separates the noise and the information of a discrete signal.
Suppose a discrete time system \[ y[n] = \sum_{k=1}^{K} a_k y[n - k] + \sum_{m=0}^{M} b_m x[n - m]\]
K y M are the order of the filter.
We must know the initial condition.
Gain
\[y[n] = G x[n]\]
Delay of \(n_0\) samples
\[y[n] = x[n - n_0]\]
Two points moving average
\[y[n] = \frac{1}{2} (x[n] + x[n - 1])\]
Euler approximation of the derivative
\[y[n] = \frac{x[n] - x[n - 1]}{T_s}\]
Averaging over N consecutive epochs of duration L
\[y[n] = \frac{1}{N} \sum_{k=0}^{N-1} x[n - kL]\]
Trapezoidal integration formula
\[y[n] = y[n - 1] + \frac{T_s}{2} (x[n] + x[n - 1])\]
Digital “leaky integrator” (First-order lowpass filter)
\[y[n] = a y[n - 1] + x[n], \quad 0 < a < 1\]
Digital resonator (Second-order system)
\[y[n] = a_1 y[n - 1] + a_2 y[n - 2] + b x[n], \quad a_1^2 + 4a_2 < 0\]
For a system’s response to be fully described by its impulse response, the system must satisfy the following key conditions.
Linearity
If the system responds to \(x_1[n]\) with \(y_1[n]\) and to \(x_2[n]\) with \(y_2[n]\), then:
\[y[n] = y_1[n] + y_2[n]\]
Homogeneity
If the input is scaled by a constant \(c\), the output is also scaled:
\[\text{If } x[n] \rightarrow y[n], \text{ then } cx[n] \rightarrow cy[n]\]
Time Invariance
A system must be time-invariant, meaning a time shift in the input causes the same shift in the output:
\[\text{If } x[n] \rightarrow y[n], \text{ then } x[n - n_0] \rightarrow y[n - n_0]\]
Causality
A causal system is one where the output at time \(n\) depends only on present and past inputs:
\[h[n] = 0 \quad \forall n < 0\]
Stability
If the impulse response does not satisfy this condition, the system may produce unbounded outputs.
\[\sum_{n=-\infty}^{\infty} |h[n]| < \infty\]
Convolution Representation
If all condition met then \[y[n] = x[n] * h[n] = \sum_{m=-\infty}^{\infty} x[m] h[n - m]\]